NASHVILLE, Tenn. — The Yankees are one of at least two teams who would like to add Kevin Youkilis to their infield.

In an effort to fill the gaping hole Alex Rodriguez’s upcoming left hip surgery created, the Yankees have offered Youkilis a one-year deal worth $12 million.

The Indians presented the Ohio native, who played at the University of Cincinnati, with a two-year contract in the area of $16 to $18 million. Cleveland manager Terry Francona was Youkilis’ manager in Boston.

The Yankees are also talking to Mark Reynolds, whom the Orioles might be bringing back after not offering him a contract.

The Yankees also have a vacancy in right field that needs to be filled by a trade or a free-agent signing.

“Youkilis fits that team. He is a no-nonsense guy whose focus is entirely on playing,’’ an NL talent evaluator said. “He has been hurt a bit lately, but he fits real well over there.’’

Youkilis played in 122 games last year when he split the season between the Red Sox and White Sox and batted a career-low .235 with 19 homers and 61 RBIs. He played in 120 games in 2011 and 102 in 2010.

People who know Youkilis, who will turn 34 in March, say he wants to play for a team where every game matters and winning isn’t a sometime thing. That would certainly give the Yankees the edge over the Indians.

However, nothing is certain. The Yankees’ early plan to fill the third-base hole was to sign free-agents Eric Chavez and Keppinger to form a platoon. Yet, they never made offers and Chavez went to Arizona (one year for $3 million) and Keppinger to the Cubs.

“I haven’t made offers to anyone who has come off the board,’’ general manager Brian Cashman said.

Despite people questioning the Yankees inactivity at the Winter Meetings that ended yesterday, Cashman defended his offseason moves so far. After saying he wasn’t disappointed by not getting the players who signed this week, Cashman admitted losing Russell Martin hurt.

Martin, the Yankees’ starting catcher the past two seasons, signed two-year deal for $17 million with the Pirates.

“We have spent money on [Hiroki] Kuroda, [Andy] Pettitte and Mariano [Rivera] and it was a lot. I don’t feel like we are not having a productive winter,’’ Cashman said on the way out of the Gaylord Opryland Hotel at the end of the meetings. “We are being deliberate and being slow. And everyone of the missed opportunities, that stuff happens.

“But when the dust settles we are going to be a team that people aren’t going to be comfortable playing. I think if we stay healthy we will be in position to take a shot at the [AL East title] and obviously something bigger after that. Having patience isn’t a bad thing.’’

Kuroda, Pettitte and Rivera signed one-year deals worth a total of $37 million.

As for needing to look at Francisco Cervelli and Austin Romine behind the plate, Cashman sounded as if he wanted to retain the catchers.

“In terms of what has happened this particular week those opportunities I am comfortable with,’’ Cashman said.

Yet, Martin split last week.

“I like Martin,’’ said Cashman, who didn’t make a formal offer to Martin and believed the result would have been different in another type of market. “I think Pittsburgh got a good one in Martin.’’